In one dimension, the inequality |x| < 1 is simple. It describes all points whose distance from zero is less than 1. On a number line, this is just the open interval (-1, 1). It is a "neighborhood" around the origin.

When you move into two dimensions, the concept of "distance" becomes more interesting. If you want to describe a region where the sum of the distances from the axes is less than a constant a, you write |x| + |y| < a. You might expect this to draw a circle, but it doesn't. Instead, it draws a perfect, tilted square—a diamond.

This page provides a widget to help you feel the boundary of this shape. You can drag a point P(x, y) around the plane to see how its "Manhattan distance" (|x| + |y|) changes, and you can adjust the constant a to see how the diamond expands or shrinks.

The widget

|x| + |y| = 0.00

The white grid represents the Cartesian plane. The thick black lines are your x and y axes. The red diamond is the boundary defined by |x| + |y| = a.

The blue dot is your point P(x, y). As you move it, the readout at the bottom updates in real-time. If the readout is blue, your point is inside the diamond (|x| + |y| < a). If it turns orange, you have stepped outside the boundary (|x| + |y| > a).

Try these

Use the widget to explore how the boundary reacts to change:

What you are actually seeing

The diamond shape isn't a coincidence; it's the result of how absolute values behave in different quadrants. To understand why the boundary |x| + |y| = a forms four straight lines, we have to look at the four quadrants of the plane separately.

In the first quadrant (x \ge 0 and y \ge 0), the absolute value signs do nothing. The equation is simply x + y = a, which is a straight line with a slope of -1.

In the second quadrant (x < 0 and y \ge 0), the x term becomes negative inside the bars: -x + y = a. This rearranges to y = x + a, a straight line with a slope of +1.

In the third quadrant (x < 0 and y < 0), both terms become negative: -x - y = a. This rearranges to y = -x - a, a line with a slope of -1.

In the fourth quadrant (x \ge 0 and y < 0), we have x - y = a, which rearranges to y = x - a, a line with a slope of +1.

When you stitch these four line segments together at the axes, they meet perfectly at the points (a,0), (0,a), (-a,0), and (0,-a), forming the diamond shape you see in the widget.

Why this matters

This concept is the foundation of what mathematicians call the L_1 norm or Manhattan distance.

In standard Euclidean geometry (the kind you use with a ruler), the distance between (0,0) and (x,y) is \sqrt{x^2 + y^2}. This is the "as the crow flies" distance. If you were to graph \sqrt{x^2 + y^2} < a, you would get a circle.

But in a city laid out in a grid (like Manhattan), you cannot walk through buildings. To get from the origin to a point (x,y), you must walk |x| units horizontally and |y| units vertically. Your total travel distance is |x| + |y|.

When we study inequalities like |x| + |y| < a, we are actually studying the "unit circle" of a different kind of geometry. In "Taxicab geometry," a circle is a diamond. Understanding this helps you realize that "distance" and "shape" are not absolute truths—they depend entirely on the rule you use to measure them. This distinction is a common theme in JEE Advanced coordinate geometry and higher-level vector analysis.

What the widget does NOT show you

While the diamond is a perfect visual for the L_1 norm, there are limits to this specific visualization:

References